Number processing in bilinguals : modelling numerical cognition : language effects on number processing

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UNIVERSITETET I OSLO

Number processing in bilinguals

Masteroppgave i spesialpedagogikk Utdanningsvitenskapelig fakultet

Institutt for spesialpedagogikk

Våren 2009

Modelling numerical cognition: Language effects on number processing

Martin M. T. Brierley

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Abstract

The goal of this literature review is to investigate if there is any reason to suspect that bilingualism in itself in can affect numerical cognition. The assumption is that this is not the case, meaning the other factors than bilingualism itself - such as language comprehension or other cultural educational or socioeconomic factors must account for any differences in mathematics performance between various bilingual and monolingual populations.

This thesis is an attempt to assess the current level of knowledge on the subject.

Various models of numerical cognition are reviewed and are subject to a theoretical discussion, in which their general merits and ability to predict language effects on number processing are considered in light of relevant research.

Existing general models of numerical cognition are not designed from the outset to accommodate, and therefore make predictions on, interaction between multiple languages and number processes. Based on the reviewed research and models, a very tentative general model of bilingual numerical cognition is outlined. Predictions on the effects of bilingualism on number processing are discussed. Pedagogical implications of bilingual numerical cognition are considered briefly. Some thoughts on the possible benefits of expanding this field of knowledge in general and for special needs

education concludes the thesis.

Contrary to expectation, available evidence suggests that language does affect numerical cognition, although the effect is limited to processing of exact numbers.

There is also evidence that aspects of number processing in bilinguals, particularly retrieval of verbal arithmetic facts such as tables memorized by rote, varies across languages. However, while these effects are important from a descriptive perspective in cognitive psychology, they are likely to be relatively insignificant from a more normative, educational perspective, as compared to other factors that affect mathematics performance in bilinguals, such as issues related to comprehension.

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Acknowledgements

Firstly, I would like to thank my mentor Guri A. Nortvedt for enthusiastic guidance, humorous scientific chit-chat and loans from her personal library. I would also like to thank my father David and my friend Knut for useful comments on language and content respectively. Last, but not least, I would like to thank Janne-Charlotte for enduring my reclusive behaviour, mood swings and erratic sleep pattern during the last two months.

Carpe Noctem!

Nesodden, May 30th, 2009 Martin Brierley

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Figures

Figure 1: The major processing components of the Modular Theory...37 Figure 2: The Multiroute Model...40 Figure 3: A simplified visualization of the Triple-Code Model...45 Figure 4: An Encoding-Complex model of number processing in Chinese-English bilinguals...54 Figure 5: A tentative, generalized version of Campbell and Epp's Encoding-Complex model...66

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1 Introduction

In Norway, as in other western countries, the debate on language issues in education continually resurfaces in the popular media, provoking no end of interest. Opinions are rife as professed and self-professed experts on the subject, teachers, politicians,

writers, organizations, parents and just about everyone else contribute to the discourse.

However, the debate seems to have two contradictory aspects to it. While both concern bilingual education, they generally appear to be considered separate issues:

On one hand there is the familiar debate on the problems that are faced by (or posed by, according to perspective) language minorities in society in general and in school in particular - and what is to be done about it (e.g. Arbeiderpartiet, 2009; Clemet, 2002;

Skaar, 2006). On the other hand, there is a debate which seems to be rooted in the need for the corporate sector to do business abroad and therefore to employ people who master additional languages, in order to remain competitive in a globalized

marketplace (e.g. Engström & Tessem, 2007; Ernes, 2007).

The European Union, while having no official educational policy of its own, nevertheless recommends that its school-age citizens should learn at least two languages in addition to their own (Espeland, 2009). Many pupils from language minorities, particularly those whose first language is not a minority language

internationally (e.g. Turkish, Arab, or Spanish), should in theory be attractive additions to the workforces of the countries in which they reside. However, the fact remains that many groups of language minority pupils do not perform adequately in school, as measured by average grades or average test results, when compared to majority pupils (Hvistendal & Roe, 2004; Marks, 2005; Rönnberg & Rönnberg, 2001).

The nature of these challenges themselves, and particularly the actions required to rectify this situation, are subjects of much controversy. One common assumption regarding language minority pupils is that their difficulties primarily due to difficulties in comprehending the majority language. Accordingly, popular school-level

interventions are diverse programs whereby the language minority pupil is either denied the use of his or her mother tongue during school hours, in order to ensure

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maximum exposure to the majority language, as well as various "softer" approaches where the language minority pupil receives various amounts of tutoring in his or her first language (Baker, 2006).

When it comes to mathematics and understanding numbers, there seems to be a popular perception that it is less dependent upon language than other school subjects and that it therefore should somehow "even the field" between minority and majority (Rönnberg & Rönnberg, 2001). However, this does not seem to be the case. Pupils from language minorities worldwide do not necessarily perform better in mathematics than in other subjects. On the contrary - quite often, mathematics seem to be one of the more challenging subjects (Marks, 2005; Rönnberg & Rönnberg, 2001).

A reason for the perception of mathematics as an "universal language" may be the reliance of mathematics on the common Arabic numeral system and international symbols (Rönnberg & Rönnberg, 2001). However, in reality Arabic numbers are nothing more than a notation system that has been accepted internationally (although hardly universally) and which has been implemented into a number of languages as the preferred system for the notation of quantity. Also, many mathematical terms are words that are "borrowed" from the "host language", but which take on a different use and meaning in the mathematical context. This jargon or "mathematic language" is often called the mathematics register (Lee, 2006; Rönnberg & Rönnberg, 2001). One characteristic of the mathematics register is that it demands a much higher level of precision than the common language, as just one misplaced word may alter the mathematical problem completely. Its use is also characterized by a lack of extra words that would otherwise be required in order to form proper sentences (Lee, 2006).

Accordingly, language must logically play an important part in number comprehension and mathemathics education.

When observing the considerable controversy around the subject of educating pupils who use multiple languages, it is tempting to ask if this controversy may be due lack of knowledge. Is our understanding of the mechanisms underlying bilingualism

inadequate, giving birth to unfounded assumptions and misconceptions? If some more basic knowledge of the invidual cognitive components of bilingualism was attained, could this clarify some basic premises and thereby lead to a more fruitful debate on the

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issue - and eventually to better educational models?

When specific groups of pupils exhibit problems in adapting to mainstream schooling, it often becomes the task of the special needs educator to examine the individual situation and to attempt to rectify the problem. Special needs educators should therefore expect to come into contact with issues related to minorities and

bilingualism. Accordingly, the field of special needs education in Norway has been expanded to incorporate research on this issue (Lie, 2004).

The decision was therefore made to examine one particular aspect of the relationship between bilingualism and mathematics. In keeping with the idea of examining as basic factors as possible, it was decided to focus on if and how language affects use and manipulation of number. If it should prove to be the case that language does affect number processing, it would imply that monolinguals and bilinguals are cognitively different.

Most of the literature on language effects on mathematics education deals with monolingual or bilingual pupils' comprehension of the mathematics register, the structure of mathematical problems and what can be done to facilitate it (e.g. Barwell, 2005; Lee, 2006; Miura & Okamoto, 2003; Rönnberg & Rönnberg, 2001). As little learning can take without language comprehension, it is only natural to assume that this is the major factor in the relationship between language and mathematics. Indeed, it would be reasonable to assume that it can account for all of the difference between bilinguals and monolinguals that can reasonably be attributed to language factors and that no cognitive difference exists between the group.

The questions that form the basis for this thesis are therefore: To what extent does language affect number processing? Are there any potential differences in bilingual numerical cognition and if so, are there any implications for mathematics education of bilinguals? The assumption is that there are no differences amongst bilinguals and monolinguals and therefore that only external factors such as comprehension of the teaching language and the mathematics register affects bilingual numerical cognition.

How can one investigate these issues? A complicating factor is that the mind is a complex and poorly understood aspect of humanity (Uttal, 2005). Fortunately, over the

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last 30 some models and theorethical frameworks have emerged from the field of cognitive psychology, which attempt to model the interaction of different processes related to numerical cognition (e.g. Campbell & Clark, 1992; Campbell & Epp, 2004;

Cipolotti & Butterworth, 1995; Dehaene, 1992; McCloskey, 1992). While theoretical models can provide a basis for understanding interaction between language and

number processing, most of these models do not explicitly consider the general impact of more than one language. Therefore, not only the question of these models' general suitability and their postulations of language effects must be considered, but also their suitability as the basis for a model that does make predictions on the interaction between multiple languages and number processes.

This thesis is a literature review which is divided into five chapters; the first being this introductory chapter. The second chapter considers both some general methodological challenges in writing literature reviews, as well as some specific methodological concerns that pertains to the topic of this thesis. While this is a subject that is seldom elaborated upon in this type of thesis, there seems to be little reason to suggest that it should therefore somehow be less important than for other types of research. The third chapter discusses some central concepts in the domain of numerical cognition and considers three central theoretical perspectives, or models, on the phenomenon. The fourth chapter considers these models' relative merits as predictors of language effects on numerical cognition. In the last chapter, language effects on numerical cognition are discussed. Also, how a very tentative model of bilingual numerical cognition can be considered, as well as the pedagogical implications of this model, as well and future directions.

In the following sub-chapters some terms that are central to this thesis will be discussed, namely the concepts of bilingualism, numerical cognition and models of cognition.

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1.1 Bilingualism

The meaning of the word is bilingual is deceptively simple: It is derived from the latin words bis linguae, meaning "two languages". It refers to someone who uses, or has the ability to use, two languages. However, it is less clear what sort of criteria that should be used in order to determine the language status of a given person: At what level of proficiency in the two languages can one be labeled a bilingual (Baker, 2006; Engen &

Kulbrandstad, 2004)?

For example, pupils in most schools in Europe are taught at least one language in addition to their own. Does that mean they are bilingual? As noted previously, the word "bilingual" can take on quite different meanings according to the context in which it is used. Technically, a child who has a Norwegian mother and an English father is bilingual. As is a child of deaf parents who both speaks Norwegian and Norwegian sign language (NTS). However, when the term is used in the popular media, it usually refers to children of immigrant parents from specific countries (Karrebæk, 2006).

The impromptu definition above, "someone who uses, or has the ability to use, two languages" is is in reality two alternate definitions: One which refers to the use of languages ("someone who uses") and one which refers to ability ("has the ability to use"). Various criteria have been suggested for both definitions. Ability-related criteria have ranged from having the ability to understand utterances in another language, to having native-like command of two languages (Engen & Kulbrandstad, 2004). Criteria for use can span from alternately using two languages (Engen & Kulbrandstad, 2004), to someone who meets and has use for two or more languages in their daily lives (Karrebæk, 2006).

However, no current definition is entirely satisfactory and to create a definition that can be used to determine exactly who is and who is not bilingual, may ultimately prove futile (Baker, 2006). In the context of this thesis, however, the bilinguals that are of interest are primarily those who command both languages well enough that

comprehension problems do not obscure any cognitive challenges related to the interaction between multiple languages and number processes.

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1.2 Numerical cognition

Numerical cognition, or how numbers are processed mentally, has been the subject of research both on humans and animals for a long time (Zorzi, Stoianov & Umiltà, 2005). There are at least three features which set numbers apart from other domains of cognition: Firstly, numbers represent a very specific aspect of reality, namely

numerosity. Secondly, numbers are the object of several other specific processes such as comparison, estimation and calculation. Thirdly, numbers can be represented in several different formats, such as different types of digit systems, written and spoken number words, etc. (Noël, 2001). The term numerosity is used to refer specifically to a measurable numerical quantity (Dehaene, 1992).

The basic understanding of magnitude and simple arithmetic ability is fundamental to humans - infants can sum and subtract some numerosities even before knowing

number words (McCrink & Wynn, 2004). Even animals, such as pigeons, can compare and perform approximate mathematical operations on sets of objects (e.g. Brannon, Wusthoff, Gallistel & Gibbon, 2001). However, there has been much controversy regarding skilled arithmetic performance in humans, especially regarding the nature of underlying mental representations of number (Zorzi et al., 2005). This issue is

obviously important to the subject of this thesis, and will therefore be discussed more thoroughly in chapter 3.

1.3 Models of cognition: Mind vs. Brain

The models discussed in this thesis mainly derive from the fields of cognitive psychology and cognitive neuropsychology. Cognitive psychology investigates the nature of the mental processes that underlie cognitive abilities, such as language understanding and production, information storage, object recognition, social

interaction, etc. (Coltheart, 2001). Cognitive neuropsychology is a branch of cognitive psychology that relies on studies of the impaired mind for the acquisition of

knowledge on the normal brain. It is not a variant of neuropsychology, because while neuropsychology is concerned with the functions of the physical brain, cognitive neuropsychology investigates the functional processes of the mind (ibid.). This distinction is important because of the mysterious nature of the connection between

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mind and brain: We know that the mind is created by the brain, but we yet to understand how (Uttal, 2005). Consequently, studies of the physical brain, while fruitful from a neurological or neuropsychological perspective, have only limited practical significance to disciplines such as psychology and educational research (Cubelli, 2009).

Studies of the mind, on the other hand, are mostly based on behavioural data and can form the basis for clinical endeavours such as rehabilitation and assessment of

practical ability (Coltheart, 2001). A couple of points should be kept in mind: Models in cognitive psychology represent systems that have no tangible substrate, they are not models of actual mechanisms. They are also incomplete representations of the systems they represent, as the human mind is much too complex to model in its entirety (Uttal, 2005). Consequently, cognitive models must reduce reality into comprehensible simplifications of very specific processes, whilst ignoring most of the factors that affect the complete system (Willingham, 2009). Further aspects of cognitive neuropsychological research and its suitability for educational inferences will be discussed in chapter 2.5.

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2 Method

This thesis is a theoretically oriented literature review. A theoretical review is,

according to Cooper (1998), a text in which one hopes to present theories that offer to explain a particular phenomenon and compares their "breadth, internal consistency, and the nature of their predictions" (p. 4). In addition, theoretical reviews often contain descriptions of experiments and may contain reformulations or integrations of notions from the theories presented (ibid).

Information on such theories and experiments must necessarily come from the existing body of scientific literature. In literature reviews, the process of searching through and selecting texts from this body can therefore be considered equivalent to the process of collecting empirical data in empirical studies (ibid.). However, the process of

reviewing literature is not exclusive to theoretical reviews. It has significant

similarities to historical research, where one gathers information from text material of the era one wishes to study. Indeed, a historic source may be defined as any source that was already in existence prior to the educational researcher's decision to utilize it. In other words, everything other than new empirical data generated during the research process can be considered to be historic sources (Fuglseth, 2006; Tveit, 2002).

Gall, Gall and Borg (2007) emphasizes that any type of educational study should contain a thorough literature review, in order to show that the author has a good overview of the field in which he is operating and that he is up to date on recent developments, so that these can be taken into consideration. Unfortunately, the literature review is often slighted - sometimes to the detriment of the value of the empirical portion of the research (ibid.). Befring (2007) notes that the methods used are rarely subject to critical discussion in educational research that is based on text analysis. He suggests that this may, at least in part, be due to a general lack of

emphasis on methodic issues in historical research - and that this is evident through the lesser amount of publications that detail principles and norms for this kind of

research, compared to the extensive body of literature on empirical research methods.

According to Fuglseth (2006), some of the more important issues in this respect are

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issues related to source evaluation and hermeneutics, or - in more common terms - assessing the quality of sources and issues related to their interpretation. Also, there is the issue of how to find and obtain access to sources of interest. In this respect, the challenges are clearly somewhat different in historical research and theoretical reviews: In the former any text is interesting as long as it can be used to shed some light on the subject of interest - indeed, historical sources need not be texts at all, although they mostly are (Tveit, 2002). In the latter, only literature adhering to strict scientific standards are of value. Therefore libraries, indexes of dissertations and theses and scientific databases become the primary points of access for the researcher (Ary, Jacobs & Razavieh, 1996).

The issues mentioned here will be examined more closely in the following sub- chapters. First, some common norms for source evaluation in historic research are discussed. Secondly some specific issues regarding the use of research reports as sources are considered. Thirdly the procedures used to find relevant sources for this thesis are explained, and thereafter the role of hermeneutics are briefly considered.

Finally some consideration will be given to an issue that is specifically related to the theories discussed within this thesis, namely the validity of inferences from studies in cognitive psychology and cognitive neuropsychology.

2.1 Source evaluation

Source evaluation is the process whereby the investigator attempts to determine how trustworthy source material is and its relevance and scientifical soundness (Befring, 2007). By its nature, historical research is one discipline that to a large extent relies upon compiling evidence from already existing source material. It is therefore only natural that this field of research is where many guiding principles of source

evaluation originates. It stands to reason that the value of a historical study primarily relies upon the researcher's ability to judge the authenticity and validity of his sources (Gall et al., 2007). The same can be said for the theoretical review. Indeed, as Mertens and McLaughlin (2004) point out, the critical analysis of sources is an important part of the research process of any literature review. Indeed, as noted earlier: Since any research paper should contain a thorough review, source evaluation is important to any

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educational research. It can be useful to divide the concept of source evaluation into two areas: Internal and external criticism (Ary et al., 1996).

2.1.1 External criticism

External criticism is leveled at circumstances pertaining to the creation of the text (Ary et al., 1996; Gall et al., 2007; Tveit, 2002). Gall et al. (2007) suggests that in order to evaluate a document on the external level, one should examine aspects such as the genuinity of the source, who the author is, where it was written, when it was written and under which conditions it was written. This is because it is in the interest of the investigator to get as close as possible to the actual event and because the

circumstances surrounding the creation of a text can affect its content: If, for example, a text was created under pressure from a third party, there is little doubt that its

contents would be radically different to what they would have been if the author created it of his own free will. One frequently used example of a situation where research may be affected by the conditions surrounding it is research that is comissioned or financed by parties that have an interest in its outcome.

Of course, not all principles of historical research are equally applicable to a

theoretical review, since the theoretical review is based on research literature, which is expected to conform to criteria that are quite different from common texts. As far as articles published in many scientific journals are concerned, there are often some safeguards already in place, e.g. a panel that will screen articles prior to publishing, in order to prevent avoid forgery, ensuring that credit is give to the correct authors, that correct bibliographical information is provided and so forth. However, multiple authors are often credited with writing a single article. This can make it difficultto determine how much of a text and which parts of a text each author is responsible for (Gall et al., 2007).

In many journals, there are also mechanisms to help the editorial staff determine the quality of submissions such as a peer-review system. In spite of such safeguards there has been examples of fraudulent research even in internationally renowned journals.

One such example is the infamous Jon Sudbø case, in which a respected name in cancer research was discovered to have fabricated all his data material (see e.g.

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Dahlberg, 2007).

While one seldom can be absolutely certain of the genuinity of a text, one can,

according to Gall et al. (2007) create hypotheses on the fraudulent nature the text and examine it with these hypotheses in mind. Viewing the text from such a perspective may lead to the discovery of information within the text that that makes the hypothesis untenable, thereby building confidence in the text. Also, the reliability of one's own text can be improved by expressing any doubts about the quality of a source (ibid.).

However, it is not hard to spot one difficulty regarding this proposition: Finding the confidence to question senior researchers in a field may prove challenging; especially to students or junior researchers.

Tveit (2002) adds that one should also consider the closeness of the source to the event it describes and the author's use of his sources. The author of the source text did, of course, face the same methodic challenges as far as souce evaluation is concerned, as the reviewer does. Sources are commonly categorized according to their distance in time and space to the event they describe. These guidelines are equally applicable for evaluating the source author's use of sources and for evaluating the importance of the source in relation to the review.

The most common way of classifying sources in in educational research it to divide sources into two categories: Primary and secondary sources (e.g. Ary et al., 1996;

Fuglseth, 2006; Gall et al., 2007; Mertens & McLaughlin, 2004). Primary sources are texts that are authored by an actual witness to an event, while secondary sources recount the event as reported in another text, which may be either a primary source, or another secondary source. While these categories may suffice for research that

primarily relies on empirical data rather than a literature review, theory-heavy texts like the present thesis may benefit from a more nuanced approach. Tveit (2002) uses the terms "førstehåndskilder" and "annenhåndskilder" (first-hand sources and second- hand sources - author's translations), to refer to the source's distance from the event it is describing; while the terms primary source and secondary source instead refer to the investigator's use of the sources. In the following, these concepts will be elaborated upon.

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A first-hand source is an actual report on an event by one or more people who were present, or in other words, where "only the mind of the observer intrudes between the original event and the investigator" (Ary et al., 1996, p. 490). In educational research this usually means a book or journal article which describes a study of, an opinion on, or a theory about a phenomenon, that is written by the actual person or persons who performed the research, formed the opinion, or conceived the theory. First-hand sources are preferable and use of such sources increases the credibility of the text which is based upon them. However, the situation may arise where the first-hand source is not available to the investigator. For instance, if one wants to refer to writings by Plato, the original documents simply do not in exist any more. However, this fact has not kept scores of researchers from writing texts based upon Plato's texts.

A second-hand source is a source that describes events not directly observed or experienced by the author, or as Ary et al. (1996) conceptualizes it, where one or more minds intrude between the observer and the investigator. A typical example of a

second-hand source in research literature is an article in which an author makes a point which is attributed to someone else. It is, however, a first-hand source concerning the author's own comments.

Even translations of texts are regarded as second-hand sources (Tveit, 2002). While Lev Vygotsky's pivotal works in developmental psychology are available in their original form, they are of little use unless one can read Russian. In a translation, the text has passed through the mind of the translator. While most translators can be trusted to try to keep as closely as possible to the original text, they still have to make a number of linguistic choices in order to create a clear text in another language:

Obviously, direct translation seldom produces acceptable text. Such decisions can affect the text in a number of ways and may therefore subtly affect the meaning conveyed by it. Still, scores of writings have been based upon Vygotsky's work,

without the authors having taken the time and effort to learn Russian. So while second- hand sources do not automatically increase the credibility of the text in which they are used, when extra thought is put into their evaluation and some consideration is put into the way they are used, they can be perfectly acceptable sources.

Primary source is a term which refers, in Tveit's (2002) terms, more to the status

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assigned to the source by its user, than to the source itself. A primary source is

considered more valuable and is therefore more central to the arguments presented in a text than a secondary source - irrespective of whether it is a first-hand or second-hand source. Ideally, a primary source is also a first-hand source. However, this need not always be the case, as with the above example: When basing one's work upon the writings of Lev Vygotsky without being able to read Russian, one must decide to use a second-hand source as a primary source. Obviously, other criteria in source

evaluation should then be considered extra carefully. Importantly, a primary source should always be the best source available to the researcher.

Secondary sources are sources deemed less valuable than primary sources and mainly used for more peripheral points in the text (Tveit, 2002). They are not

sufficient for making important points or underpinning central arguments that could endanger conclusions drawn if sources were proven to be inadequate. Since less emphasis is put on them, they need not pass as strict evaluations as primary sources.

However, such sources still can be important to an author, as their use can save a lot of time and resources, compared to identifying, locating and retrieving the primary

source for every minor point in a text.

2.1.2 Internal criticism

Internal criticism entails considering the quality of text itself (Ary et al., 1996; Gall et al., 2007; Tveit, 2002). One should consider the internal consistency of the text, if the arguments in it are logical and if there are contradictions or other elements that

subtract from the overall impression (Befring, 2007). Also, one should consider the accuracy and worth of the statements within. This makes internal criticism a somewhat more complex task than external criticism (Gall et al., 2007). There are two

overarching theories on how to should judge the truth or falsity of a statement are the correspondence theory and the coherence theory (Tveit, 2002):

Correspondence theory forms the basis of empirical research. According to this theory, a statement is true if it corresponds with reality - i.e. if it corresponds with that which is observable. Applied to research literature, this would mean that a statement is true if it corresponds to the (empirical or historical) data presented (Tveit, 2002). In

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other words, convincing arguments can be put forwards to the effect that it is backed by the evidence presented it should be assumed to be true. One can also consider the realism of the events described in a text: The likelihood that the described events could have occured given the circumstances, the apparent reliability of the information

presented, etc. (Gall et al., 2007).

Coherence theory, on the other hand, postulates that a statement is true if it is consistent with what else is known on the subject (Tveit, 2002). Applied to research literature, this means that the statement is true if it consistent with other statements or theories on the subject. A problem with the coherence theory is that if one, in an empirical study, finds something that is radically opposed to commonly accepted knowlendge, one may reject the findings, even if they are true. On the other hand, if one keeps to correspondence theory only, there is a risk of endorsing research results that are false.

As a result, it is good practice to take both theories into account, and both consider if a statement both corresponds with the data it is based upon and what else is commonly known on the subject. One should therefore attempt to consult multiple sources on the same subject. Replicated studies are an advantage in this respect, but with some types of studies, particularly case studies, replication can prove difficult. This point will be adressed in chapter 2.5, in which some types of studies incorporated into this review will be discussed.

2.2 Research reports as sources

A few issues concern articles that contain reports of empirical research which are not equally relevant relevant to other types of source texts in literature reviews or

historical studies. Since a large part of the sources for this thesis are research reports, this sub-chapter is devoted to such issues.

One issue is that one must not only consider the quality and origins of the text itself, as described in the previous sub-chapters. It is also necessary to judge the quality of the research that it describes (Gall et al., 2007). To this end, one must examine the text for descriptions on how the research procedures were carried out. Something about the

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preparatory work is often mentioned in the introduction to a report, most reports will also have a method chapter describing the procedures that were followed. However, there are often restrictions on report lengths in journals, which can lead to the omission of details that would have been useful in evaluating the validity of the research

presented, which can cause the reader to have to make assumptions or guesses about aspects of the study (ibid.).

Mertens & McLaughlin (2004) suggest a number of critical questions that can help in analyzing the data analysis and interpretation of data in empirical studies (see pp.

201-202). These will not be listed in detail here. However, the central points concern such factors as evaluation of the statistical procedures used in quantitative studies, the regularity of the data and the degree to which peer researchers have been consulted for alternative perspectives on its interpretation in quantitative studies, and if alternative explanations are accounted for.

Another issue pertaining to research reviews is the degree to which the studies that are reviewed are representative of studies on the subject in general (Cooper, 1998).

Particularly in well-known international journals access to publication of papers is tightly regulated. Therefore, a great deal of the research done is never published in such journals, making it difficult to access for the investigator. There may be features common to published reports that are less common in research in general. For example it is not likely that research projects in general produce statistically significant results as frequently as indicated by the number of such results presented in journal articles (ibid.). In addition, investigators usually only have access to a limited number of retrieval channels, which may skewer the selection of research reports (ibid.).

All in all, it is improbable that research retrieved for a review is going to be

representative of all research conducted on the subject. Cooper therefore suggests that when reviewing sources, the investigator should try to search as broadly as possible and should attempt to imagine what inaccessible studies might have said and how they might differ from the ones retrieved. Naturally, this is not something which is easily accomplished. However, most importantly, Cooper argues that the investigator should keep account of the search process and that the review should feature this account.

This makes the search process open to criticism and replication by the reader, in the

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same manner that research procedures should be laid open to criticism and replication in empirical research reports. As a consequence, this process is the subject of the next sub-chapter.

2.3 Literature search

The first challenge facing the prospective author of a scientific text is to get an

overview of the status in the field in which he is interested. What is the current level of knowledge on the issue? What has already been said on the particular subject (Gall et al., 2007)? Even determining where to start can sometimes be a daunting task, but fortunately there are publications that can provide the needed starting points, in the form of handbooks and other collections of writings that provide overviews of the field, reviews of trends and disscussions in the field and research syntheses (Ary et al., 1996). Such sources can be termed "preliminary sources" (Gall et al., 2007). Examples of such books used during the work on this thesis are the Handbook of Child

Psychology, sixth edition (edited by Damon & Lerner, published in 2006 by John Wiley & sons, Hoboken) and The Handbook of Mathematical Cognition (edited by Campbell, published in 2005 by Psychology Press, Hove). While such compilations may be a good starting point, they rarely provide insight into the very latest

developments in the field. Even if they are quite recent, the process of compiling, publishing and distributing such a book takes time, and so a full overview can not be made without extensive forays into the world of scientific journals.

Gathering journal articles can be an exhausting process, although the event of computerized databases have made this task somewhat easier than it used to be (Mertens & McLaughlin, 2004). The event of digital full-text articles has further improved it. Recent development in the field continue the trend of making scientific literature more accessible and to this end the library at the University of Oslo provides the X-port service (http://x-port.uio.no/), which is a search engine capable of

performing meta-searches in a number of scientific databases.

A meta-search is a search that is conducted in several databases simultanously or in rapid succession without requiring further input from the user, greatly reducing the amount of labour involved. By selecting a X-port category such as

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"utdanningsvitenskap" (educational research) or "psykologi" (psychology) and

"fagdatabaser" (scientific databases), one can select any number of databases from a list and the system will search them according to the specified keywords, all the while marking articles which are available in full text to be downloaded, or providing library data information for those that are not.

This convenient service has been used extensively during the research for this thesis, both initially, as a tool to get an overview and later, in order to locate articles on a specific subject, search for other works by specific authors or groups of authors, or find articles referred to by other authors. Its coverage is extensive and includes

numerous databases such as ERIC, PsychINFO, EMBASE and MEDLINE. In order to test its coverage, searches were conducted directly in some databases such as ProQuest Psychology and the ISI Web of Science. However, these did not turn up results that were not available through X-port as well. Also, during work on this thesis, X-port never failed to locate a specific article when queried. This suggests that it does cover the field well enough for information retrieval for a project such as this thesis. It was also noted that articles often are present in more than one database anyway, especially if they have been influential in the field or are written by renowned authors.

The majority of the initial database searches were necessarily of a general nature, the aim being to get an overview of recent developments in the field. Therefore, the most important keywords used were language-related ("language", "speech", "bilingual", etc.), maths-related ("maths", "mathematics", "numeracy", etc.) and brain-related ("brain", "mind", "cognition", etc.). Since the aim was to get an overview of current status in the field, a time frame of 15 years was initially decided upon. Later on, as the search progressed towards more specific subjects, the time frame was expanded, so as to also be able to find older, but important articles.

After obtaining search results, abstracts and bibliographical data were retrieved for the more interesting-looking articles, in order to perform rudimentary quality control by considering such factors as the number of times they had been cited, what journal they had been published in, who their authors were and which institutions their authors were associated with. Fairly quickly some author names started to appear more often than others, suggesting that some researchers whose fields of interest lay particularly

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close to the subject of this thesis, whose names could then be looked up in journal databases, library databases, and on the Internet in general. Often, researchers have their own homepages, on they explain their fields of interest, place of work and list their bibliographies. Such homepages can therefore prove quite useful in finding information on the author, locating additional work, other authors in the same sphere of interest, etc.

When conducting meta-searches, limiting the searches is paramount. Otherwise one may get an unwieldy number of search results. One way of limiting the searches in scope is to limit the type of articles requested. Research reports were, of course, included. As were articles that presented theoretical models based on empirical evidence. Articles that commented on, or brought new perspectives, to other research reports were also included as the debate following the publication of controversial articles can be very interesting from a theoretical point of view.

Naturally, as work progressed searches became increasingly specific, due to the fact that one develops an increasingly clear idea of what one needs of further information:

Either in terms of writings by specific author, a specific subject, perspective - or a book or article that has been referred to in another text which needs to be investigated further for purposes of verification of a statement, or to provide additional information.

2.4 Hermeneutic considerations

When an article, book chapter, or other text has been located, evaluated, judged to be relevant and trustworthy and its potential use determined, the only matter remaining is how the content of text is interpreted by the investigator. This subject is the matter of this sub-chapter.

The term "hermeneutics" originated in theology, as term decribing techniques for an important endeavour in that particular field, namely the interpretation of bible texts (Hjardemaal, 2002). Since, the term has been adopted by other frequent interpreters of text, such as historians and law researchers and expanded to mean the interpretation of a text in general (Befring, 2007). The process of text interpretation and its use in a new text can be divided into three stages: Firstly the semantic meaning of the words are

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decoded. Then one attempts to extract the general meaning, or message of the text as a whole. Finally one expresses the general meaning in one's own words, in creating the new text (Kjeldstadli, 1999). The first step of the process merely requires command of the language in which the text is written, so that meaning can be attached to the

writing. Apart from any language-based misunderstandings that may occur at this stage, it is mainly in the second and third step that the original message is prone to distortion. It is therefore good reason to be mindful of factors that may colour one's interpretations of the sources.

Many of these factors are related to what is often termed preconceptions (Alvesson &

Sköldberg, 1994; Wormnæs, 2006). Preconceptions are based on our earlier experiences and serve as "spectacles" through which reality is observed. They are useful because they allow us to quickly assess everyday information based upon previous experience. However, they are less useful when trying to view information objectively, since they are not necessarily rational, or applicable to all contexts. If, for example, one is observing a child solving an arithmetic task and making off-hand assumptions about the reasoning behind a particular strategy used by the child, these may prove to be quite different from the child's reasoning, as described by the child itself.

The process of text interpretation is often described as a circle, wherein the reader gains an increasingly good understanding of smaller parts of the larger message. This in turn resulting in better understanding of the meaning of the whole message, which in turn leads to an even better understanding of the part. This is often termed the

"hermeneutic circle". For each transfer from part to whole and vice versa, the understanding passes through, is modified by, and may in turn slightly modify the preconceptions involved (Alvesson & Sköldberg, 1994).

How we perceive the language of a text, as well as our feelings, mood, wishes,

interests, values, attitudes and theoretical perspectives all contribute to form the basis for our understanding of a text (Wormnæs, 2006). Wile it is neither realistic nor desirable to rid oneself of all prior influence, Alvesson and Sköldberg (1994) notes that reflecting upon one's preconceptions in a systematic way can go a long way in improving the validity of interpretations. One should therefore attempt to clarify which

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of one's beliefs, values and experiences that are relevant to the topic being

investigated, as being conscious of one's preconceptions may negate some of their impact on the interpretation of information (Gall et al., 2007).

2.5 Validity of inferences in cognitive psychology

The data from which models in cognitive psychology are constructed mostly originate from studies of cognitively normal people who perform a task such as solving an arithmetic problem (Coltheart, 2001). Another important source of data is studies in cognitive neuropsychology investigating people who are cognitively impaired, due to e.g. brain damage, disease or disability (McCloskey, 1992). In both cases, inferences are based upon measured factors such as the error rates or error types that subjects produce during completion of tasks, differences in completion speeds, imagery from brain imaging techniques, or behaviour that the participants exhibit during task resolution. A few points concerning such inferences will now be discussed.

Single-patient studies have undoubtedly provided useful information in a variety of scientific disciplines. Such studies can be especially useful when the individual studied exhibits unusual traits (Befring, 2007). This type of study is quite common in

cognitive neuropsychology, as one investigates the impairment of a brain damaged or otherwise cognitively impaired person. While a description of an impairment can be interesting by itself, it is not uncommon for researchers in cognitive neuropsychology to attempt to extrapolate from their findings onto the general, cognitively normal population. Such inferences are evident in many cognitive neuropsychological studies (e.g. Cipolotti & Butterworth, 1995; McCloskey, Sokol & Goodman, 1986), which raises the question: Given that every human is an individual, how can one claim that one human can represent all of humankind in matters of cognition?

The basis for this endeavour is the assumption that architectures of cognition are universal (Coltheart, 2001). Another example of such logic can be found in the field of medicine: Since humans for the most part are anatomically similar to each other, a bone found in one human can generally be expected to appear in all other humans.

However, since cognitive mechanisms are not directly observable, inferences must be based on the observation of behaviour that is somehow qualitatively different from

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similar behaviour in the cognitively normal. The assumption is then that cognitive impairments can reveal features of the normal mind by way of their absence. The psychological term for such an inference is a dissociation (Coltheart, 2001 ;

McCloskey, 1992). However, in order to strengthen the case for such an inference, one should ideally be able to also exhibit the opposite case, thereby making it a double dissociation (Noël, 2001). As an illustration of this logic one can imagine a stereo system that ceases to play music through the left speaker. If the speaker itself and all the cables are intact, this would suggest that inside the system there are separate circuits for left and right sound production. If one subsequently came across a stereo system that would play through the lesft, but not the right speaker, this would be a double dissociation, and would make a stronger case for the inference.

Another point concerning single-case studies is the question of replication. Since the most interesting cases often will be the most unusual ones, it goes without saying that replicating such a study can be quite impossible. If a study is done on a patient with a one-in-a-million injury, it is unlikely for other researchers to come across a similar case. If the investigator doubts a finding in such a case, there is in reality only two possibilities that are open: Either to reject the assumption of the universality of

cognitive processes or reject the research methods used (Coltheart, 2001). It therefore falls upon the original researcher to conduct as sound an investigation as possible, so as to minimize the possibility of errors in the study.

Group studies are common in cognitive psychology, the rationale being that one can infer something about underlying cognitive processes by observing behaviour in cognitively normal subjects (Coltheart, 2001). Of course this requires studying more than a few subjects in order to eliminate interference from abnormal subjects,

strengthening the case for extrapolating the results to the people in general. However, one notable feature of such studies is that the number of participants often is relatively small (see e.g. Campbell & Xue, 2001; Spelke & Tsivkin, 2001). One reason for this may be the fact that to be able to extract information of the necessary detail level, one must often have subjects complete fairly large sets of tasks, so that the necessary level of measurement sensitivity is achieved. This does, of coures, increase the cost of the study in time and financial resources, thereby limiting its size. Recently, brain imaging

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techniques have also been employed in group studies, in order to make inferences not only on the basis of behaviour, but also on brain activation. The employment of such techniques also increases complexity of the investigation and the cost per subject further.

Commonly, results are (often implicitly) generalized to a very large population. For example a study of language switching effects on arithmetic involving 26 Chinese- English bilingual students might bear a title such as "language switching in Chinese- English bilinguals" (example from Campbell, Kanz & Xue, 1999), implying

generalization to all Chinese-English bilinguals on the planet, regardless of languages known, age and education level. However, the same logic apply to this type of studies as with single patient studies - the assumption that cognitive structures are in principle universal. Also, contrary to single-case studies, replication of group studies is quite possible if they are well conducted and described. Therefore, studies that fully or in part replicate such a study or otherwise supports its results can strengthen the case for the validity of inferences.

Studies using brain imaging techniques face their own set of validity issues,

according to the technical limitations of different imaging technologies and how image results can reasonably be interpreted. All studies that employ brain imaging techniques that are referenced to in this thesis used functional Magnetic Resonance Imaging (fMRI) technology. The purpose of fMRI is to allow one to localize brain activity (Desmond & Chen, 2002). The main reason for conducting fMRI studies in

neuropsychology is to examine how different tasks affect brain activity. While fMRI, for this purpose, is a great improvement over previously available techniques of brain imaging, it still has shortcomings that may affect validity. Since only a part of the foundation for this thesis rests upon such studies, just a few points related to them will be discussed. For a review of additional factors that may affect the validity of

inferences from fMRI images, see Desmond and Chen (2002).

One technically limiting factor is resolution. For each scan, the operator has to set the required resolution, both in time and space. fMRI works by taking a series of two- dimensional snapshots of the brain, that contrasts blood oxygen levels, the logic being that increased brain activity leads to greater demand for oxygenated blood (Desmond

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& Chen, 2002). The snapshots are horizontal and vertical "slices" of the brain, in effect creating a three-dimensional grid. The spacing between these slices can be anywhere from 5-7 mm in the brain in general and 1-4 mm in designated areas. Time between snapshots can from 1-5 seconds (Desmond & Chen, 2002). Needless to say, as thoughts move through the brain's millions of cells at ligthning speeds, much can happen in the space of a few seconds - or in between two slices. This limits how

specific information that can be gleaned from such a study and thereby the precision of interpretations.

A factor that potentially affects interpretation is deactivation. In order to determine which areas are activated during a particular task, it is necessary to btain a baseline reading from the subject so as to be able to compare the "normal" activation patterns to the activation patterns during the tasks (Venkatraman, Siong, Chee & Ansari, 2006).

The commonly accepted interpretation of increased brain activity in an area, is that this area plays a part in performing the cognitive task at hand. However, researchers often also observe deactivation in some areas, compared to the baseline - an occurence that is not well understood (Raichle et al., 2001; see also Gusnard & Raichle, 2001). As a consequence of this, observed deactivations are not often discussed in the research report (Venkatraman et al., 2006). This is questionable, as even if it should be shown to have no impact on the validity of the study, it may conceal findings that could turn out to be important at a later date.

2.5.1 Neuroscience and education

As noted in the previous sub-chapter, this thesis in special needs education is based on cognitive psychology and cognitive neuropsychology. Consequently a question that needs to be addressed is: Is this a useful combination? Can and should fields related to cognitive science, and particularly to neuroscience, influence educational research in general and special needs education in particular?

According to Willingham (2009), this question is pointless, because even neuroscience has already influenced education at various points in recent history. One of the

examples used by Willingham to underline this point is particularly relevant to special needs education: Fairly recent neuropsychological research into the structures that

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support phonology in the brain, has supported the theory that dyslexia is primarily a phonological disorder, rather than a visual perceptual disorder. This was an ongoing debate in educational research to which neuropsychology contributed towards a resolution. Willingham does, however, point out some challenges to the relationship between these fields of research. He has named these challenges the goals problem, the vertical problem and the horizontal problem.

The goals problem arises from the differing goals of the two scientific disciplines. As with most natural sciences, neuroscience is descriptive, while educational research, like many social sciences, is normative in nature. Neuroscience will therefore never be able to provide the prescriptive solutions requested by educational researchers, which limits the utility of cooperation (Willingham, 2009). To counter this point, one could argue that while neuroscience by necessity is descriptive and while educational research's strong connection to the profession of teaching (with all its political saliency) means that normative research results are often expected from it, this does not necessarily mean that all educational research must be normative, nor that

educational research could not in any way benefit from research of a descriptive nature - either in educational research itself, in cognitive psychology or neuropsychology, or in other fields. Also, symbiotic relationships between descriptive and normative diesciplines are not uncommon, the relationship between physics and engineering being but one example.

The vertical problem refers to the difference in levels of analysis used in the two fields. Willingham argues that neuroscience's detailed approach to the study of

cognition means specific functions and processes are studied in isolation, disregarding as many factors as possible, so as to keep models simple enough to comprehend. The highest level of analysis in this field is therefore the interaction of function or the mapping of functions onto brain areas. In educational research, the very lowest level of analysis is the complete individual and analysis can be carried out on significantly higher levels; such as school, community, or country. Accordingly, one could argue that not only do analysis levels not overlap, there is quite some distance between the lowest analysis level of educational research and the highest analysis level of

neuroscience.

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A few counterarguments to this point can be made. Firstly, the lowest level of

educational research is not necessarily the complete human individual. Indeed, a fair amount of research is concerned with lower levels of analysis, such as individual strategies or particular behavioural features, such as subvocal speech (e.g. Corkum, Humphries, Mullane & Theriault, 2008; Ostad & Sørensen, 2007). In order for such efforts to be feasible, it is necessary to limit the number of factors taken into account.

This does not automatically make such research pedagogically irrelevant.

Secondly, the continuing specialization in and subdivision of research fields,

especially within educational psychology and special needs education, is also lowering levels of analysis. As researchers in special needs education examine ever more

specific and narrowly defined disorders, interfacing with neuropsychology can be expected to become increasingly fruitful. One example is the very pedagogically relevant educational research conducted into specific language impairment (SLI) and its subcategories (se e.g. Leonard, 1998), to which neuroscience has provided, and quite likely will continue to provide, valuable information.

The horizontal problem concerns the utility of neurological knowledge for educational purposes. As Willingham states, knowledge that e.g. the intraparietal sulcus contributes to number sense is not very relevant to educators and may even be misleading. The latter point is underlined by Mason (2009), who looks with concern upon the number of "neuromyths" that are prevalent amongst teachers, partly due to the misinterpreted neurological concepts propagated by the commercial, non-scientific

"brain-based learning" industry. However, Cubelli (2009) points out that it is not necessarily the theories on the physical brain, nor neuropsychological theories on mappings between the physical brain and cognitive functions that are most relevant to education, but the theories of mind that are derived from e.g. cognitive psychology and neuropsychology. These overarching theories of cognitive functions should, Cubelli notes, contribute to improving practice and to increase the effectiveness of the teaching experience.

All in all, one should not blindly expect neuroscience to provide the answers (Willingham, 2009). But neither should brain-related research be disregarded as irrelevant - especially fields such as cognitive neuropsychology can have profound

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implications for education, rehabilitation, treatment and assessment of people with various disorders (Coltheart, 2001; Willingham, 2009).

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3 Numbers and the mind

In the following sub-chapters, some important terms relating to numerical cognition will be discussed, followed by reviews of the three dominating models (or theoretical frameworks) for understanding human numerical cognition. The aim is to evaluate current theoretical platforms for use as a foundation on which to base hypotheses on interaction between language and number in bilingual number processing.

The process by which children come to learn number concepts and mathematical understanding has been a hotly debated issue through much of the history of research on this issue (Geary, 2006). For several decades the Piagetian, constructivist view was the dominating force in developmental psychology (Newcombe, 2002). According to this view, children construct the cognitive mechanisms needed for cognitive tasks from their experiences and so no level of inherent, biological mechanisms to facilitate such knowledge is expected to be present at the moment of birth (Geary, 2006). That a view that leaves such ample grounds for optimism regarding what can be achieved through educational means has had such a large impact on educational thinking is hardly surprising.

This is not so much the case with the rival, nativist view. In the strictest sense such a view postulates that all behaviour observed in a species must have a basis in its biological features. Since complex cognitive processes such as arithmetic and

language are unique characteristics of humankind, they must be handled by biological neural components that are unique to humankind (Newcombe, 2002). If this were true, it could mean that children with difficulties in solving arithmetic problems might have such problems because parts of their brain was underderdeveloped. The implications for education would leave little reason for optimism: Changing innate biological features is not normally within the power of educators, so at most one could hope to teach children with such deficiencies some alternative coping strategies or techniques that might limit the impact of it upon the child's future life.

However, neither nativism or constructivism in the purest sense seems to be fully compatible with a number of findings in cognitive research. Meaningful theoretical

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frameworks for understanding complex processes in the human mind must therefore be found somewhere in between these two extremes (Newcombe, 2002). It must be accepted that both experience and evolved mechanisms motivate and drive human behaviour (Uttal, 2005).

It is tempting to assume that a rigid, nativist view underlies much of the increasing amount of cognitive research that is conducted, as the search for connections between brain and behaviour continues, fuelled by technological advances that allow it to be pursued by ever more sophisticated means. Geary (2006) categorizes several notable names in psychological research, such as Brian Butterworth and Stanislas Dehaene as neo-nativists, as they attempt to tie aspects of number processing to specific locations in the human brain (see e.g. Butterworth, 1999; Dehaene, 1997). However, neither author proposes that mathematical understanding is "pre-programmed" into the brain, although both present compelling arguments to the effect that some basic processes involved in numerical cognition may be innate - and quite possibly not restricted to humans. However, Dehaene (1997) remarks that higher level cognitive functions, such as reading and mathematics, were developed too recently in human history in order for us to have evolved specific brain areas devoted to them. Some of the basics of number processing will be considered in the following sub-chapters.

3.1 Components of number processing

In order to create a hypothesis on the cognitive architecture underlying a higher-level cognitive process, one must first break the larger process down to what one believes to be the individual stages that make up that process. According to Campbell & Epp (2005), the act of solving a simple arithmetic problem can for instance be broken down into the following overarching stages:

1. Converting the stimuli into appropriate internal codes 2. Retrieving or calculating the answer

3. Producing the answer

These stages are, of course, over-simplified - but will suffice as an example for now.

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With respect to the first stage, stimuli (in the form of numerosity) can be produced and received in a number of formats. Those that have garnered most interest from

researchers are also the ones that are most language-related: Spoken words, written words and written numbers. In the first case, numbers are perceived audibly, while in the other two they are perceived visually. The format of the numbers in the first two cases fall within the category of number words (Brysbaert, 2005), as they form part of our normal vocabulary of words. Written numbers can be termed visual Arabic

numbers (Dehaene, 1992) - a term inspired by the Hindu-Arabic origin of the numeral system. They are distinct from number words because they are not connected to a specific language, and so they do not have their own verbal equivalents - each

language associates them with their own number words. Numerosities can, of course, also be perceived by other means, such as:

● Visual magnitudes, e.g. a number of objects

● Audible signals, e.g. three knocks at the door

● Tactile information, e.g. feeling a number of objects

As will become clear in the course of this thesis, the visual magnitude format has also been important to the study of numerical cognition. The latter two have, however, not garnered much interest. In a school setting, an arithmetic task given to a pupil will usually involve stimuli in the language-related formats. If, for example, the task is given in Arabic numeral format, and the teacher demands a verbal answer, the processing stages commence with the conversion of the stimuli into mental representations of numerosity, or codes. The concept of codes, or internal, mental representations of numerosities is fundamental to many theories of numerical cognition and much of the debate surrounding models of numerical cognition has revolved around the number and nature of such codes (for reviews, see e.g. Campbell

& Epp, 2005; Noël, 2001).

The three stages outlined above imply three sets of cognitive systems: One set for encoding the perceived magnitudes into the appropriate internal code or codes, one set for the retrieval or computation of the answer and of course one set that mirrors the first set, which converts numerosities from the codes back to responses which allow us

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to produce the answer in any format we know (Campbell & Epp, 2005).

An example will illustrate this better: If a teacher gives a pupil a task, for instance the arithmetic problem "3 x 7 = " presented on the blackboard as the expression and requests an oral answer to that task, the process could be divided into the following three steps:

1. The pupil reads the problem on the blackboard. The visual Arabic number stimuli is then converted to an internal code in the brain so that it can be accessed by the calculation processes.

2. The problem can now either be calculated, or the answer retrieved if it was previously learned by rote. In this example, we will assume that the pupil is familiar with multiplication tables. Therefore the answer, 21, is retrieved rather than calculated anew.

3. The answer is converted from the an internal code to verbal output, upon which the pupil speaks the answer out loud.

While this example illustrates the logic of building a model, a proper model of

numerical cognition would, of course, have to be able to make specific predictions on the nature of internal codes and how individual subsystems relate to these and

communicate with one another. The three overarching stages must therefore be broke down to a number of specific processes and steps which can explain findings in behavioral and other studies of how humans use and manipulate numerosities.

Next, some basic number processes are discussed, as these concepts are important to the discussion in the following chapters. Firstly a group of basic processes that can be termed quantification processes are considered and thereafter the concept of arithmetic facts storage and retrieval.

Number processing in bilinguals : modelling numerical cognition : language effects on number processing

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